Related papers: Nonlinear approximation of high-dimensional anisot…

We propose to approximate the conditional expectation of a spatial random variable given its nearest-neighbour observations by an additive function. The setting is meaningful in practice and requires no unilateral ordering. It is capable of…

Given sufficiently many components, it is often cited that finite mixture models can approximate any other probability density function (pdf) to an arbitrary degree of accuracy. Unfortunately, the nature of this approximation result is…

In this paper, we study functional approximations where we choose the so-called radial basis function method and more specifically, quasi-interpolation. From the various available approaches to the latter, we form new quasi-Lagrange…

We consider the problem of numerically approximating the solutions to an elliptic partial differential equation (PDE) for which the boundary conditions are lacking. To alleviate this missing information, we assume to be given measurement…

In [DL] systems of differential polynomials are investigated with respect to properties of Artin approximation type. The key tool in [DL] is an extended version of a formula by Hurwitz [Hu] expressing high order derivatives of an expansion…

We introduce a practical method to enforce partial differential equation (PDE) constraints for functions defined by neural networks (NNs), with a high degree of accuracy and up to a desired tolerance. We develop a differentiable…

It is often useful to compactly summarize important properties of model parameters and training data so that they can be used later without storing and/or iterating over the entire dataset. As a specific case, we consider estimating the…

Reduced basis methods for approximating the solutions of parameter-dependant partial differential equations (PDEs) are based on learning the structure of the set of solutions - seen as a manifold ${\mathcal S}$ in some functional space -…

The past decade has seen increasing interest in applying Deep Learning (DL) to Computational Science and Engineering (CSE). Driven by impressive results in applications such as computer vision, Uncertainty Quantification (UQ), genetics,…

While there are convergence guarantees for temporal difference (TD) learning when using linear function approximators, the situation for nonlinear models is far less understood, and divergent examples are known. Here we take a first step…

We develop a nonparametric approach to identify and estimate consumer preferences and unobserved heterogeneity under nonlinear price schedules. Leveraging variation across multiple price schedules, we show that both the utility function and…

We refine a recently-proposed class of local entropic loss functions by restricting the smoothening regularization to only a subset of weights. The new loss functions are referred to as partial local entropies. They can adapt to the…

We describe an efficient domain decomposition-based framework for nonlinear multiscale PDE problems. The framework is inspired by manifold learning techniques and exploits the tangent spaces spanned by the nearest neighbors to compress…

This paper introduces a new method for semi-supervised learning on high dimensional nonlinear manifolds, which includes a phase of unsupervised basis learning and a phase of supervised function learning. The learned bases provide a set of…

This paper tackles the issue of real-time parametric estimation of a wide class of probability density functions from limited datasets. This type of estimation addresses recent applications that require joint sensing and actuation. The…

Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…

A new approximation format for solutions of partial differential equations depending on infinitely many parameters is introduced. By combining low-rank tensor approximation in a selected subset of variables with a sparse polynomial…

The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the…

We propose a differentiable nonparametric algorithm, the Delaunay triangulation learner (DTL), to solve the functional approximation problem on the basis of a $p$-dimensional feature space. By conducting the Delaunay triangulation algorithm…

We investigate anisotropic (piecewise) polynomial approximation of functions in Lebesgue spaces as well as anisotropic Besov spaces. For this purpose we study temporal and spacial moduli of smoothness and their properties. In particular, we…